What should come in the place of question mark (?) in the following questions ?
`(16 xx 32)/(9 xx 27 xx 81)`= ?

Options

A. `((2)/(3))^(12)`
B. `((2)/(3))^(11)`
C. `((2)/(3))^(13)`
D. `((2)/(3))^(9)`

Answer

To solve the expression \((16 \times 32) / (9 \times 27 \times 81)\), we will break down each number into its prime factorization and simplify the expression step by step.

### Step 1: Factorize the numbers
- **16** can be expressed as \(2^4\).
- **32** can be expressed as \(2^5\).
- **9** can be expressed as \(3^2\).
- **27** can be expressed as \(3^3\).
- **81** can be expressed as \(3^4\).

### Step 2: Substitute the factorizations into the expression
Now we can rewrite the expression using these factorizations:
\[ \frac{16 \times 32}{9 \times 27 \times 81} = \frac{(2^4) \times (2^5)}{(3^2) \times (3^3) \times (3^4)} \]

### Step 3: Simplify the numerator
In the numerator, we can combine the powers of 2:
\[ 2^4 \times 2^5 = 2^{4+5} = 2^9 \]

### Step 4: Simplify the denominator
In the denominator, we can combine the powers of 3:
\[ 3^2 \times 3^3 \times 3^4 = 3^{2+3+4} = 3^9 \]

### Step 5: Rewrite the expression
Now we can rewrite the expression as:
\[ \frac{2^9}{3^9} \]

### Step 6: Express as a single fraction
This can be expressed as:
\[ \left(\frac{2}{3}\right)^9 \]

### Conclusion
Thus, the final answer is:
\[ \left(\frac{2}{3}\right)^{9} \]